Universe is everything that exists, it is the spacetime and everything in it, all matter and energy as well as all the laws of nature by which it behaves. The size of the whole universe is not known, it is possibly infinite, however the size of the observable universe -- the part of it with which we can ever interact due to limited speed of light combined with constant expansion of space -- is about 93 billion light years in diameter, and contains an estimated number of 100 billion galaxies, each containing hundreds of billions of stars. Current science says the universe is about 13.7 billion years old and that it started with the Big Bang, a point in time from which everything began to rapidly expand from a single point in space.
Computers can be used to simulate certain parts of the universe, in fact all programs mimic the universe in some kind of simplified way, be it scientific simulations of planet collisions, government databases or games -- they all more or less accurately model the reality. We can also see computers as a way of creating smaller universes, which leads many to think our universe may itself be a simulation running on some computer in a "bigger" universe (note that this probably isn't testable and the debate isn't scientific, but we can lead philosophical discussions about it).
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